t. odagaki and t. ekimoto department of physics, kyushu university ngai fest september 16, 2006
TRANSCRIPT
![Page 1: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/1.jpg)
T. Odagaki and T. EkimotoDepartment of Physics, Kyushu University
Ngai Fest September 16, 2006
![Page 2: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/2.jpg)
Free Energy Landscape
Order parameter
High T
Low T
Phase transition
Fre
e en
ergy
Phase Transition
Configuration
High T
Fre
e en
ergy
Glass transition
Low T
: Diverging mean waiting time gT
![Page 3: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/3.jpg)
PhenomenologyPhenomenology
Fundamental TheoryFundamental Theory
Dynamics
Single particle: Gaussian to non-Gaussian transitionSlow and fast relaxations
Specific heat: Annealed to quenched transition
Thermodynamics
Cooling-rate dependence
Construction of free energy landscape
Dynamics
Thermodynamics
Slow and fast relaxations
![Page 4: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/4.jpg)
Separation of time scalesSeparation of time scales
Total
Microscopic Relaxation
![Page 5: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/5.jpg)
Free energy landscape
])(exp[})({ 2 i
iiii RrCRr For practical calculation
dRrrHN
RNVTZ iiii })({})]({exp[!
1}){,,,(
}){,,,(ln}){,,,( iBi RNVTZTkRNVT
Dynamics on the FEL
)(})({ tQRdt
dRiiR
ii
)(tQi : Random force
[Ansatz]
)()( tuRtr iii ttt 0for )(trR ii where
tt 0Separation of time scales
![Page 6: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/6.jpg)
)()sin(
)( tQdx
xdTg
dt
dx
0)( tQ )(4)()( 210
21 ttT
TtQtQ
BkT /20
Dynamics
random force
and
A toy model for the dynamics on the FEL
Scaled equation
)()cos()( tQxTgdt
dx
)(2 Tg
![Page 7: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/7.jpg)
)(Tg
0/TT
Three models for g(T)
TT /0
)]1tanh(1/[)]/tanh(1[ 0 TT
movie
1)( Tg
![Page 8: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/8.jpg)
The dynamical structure factor of Model 1
0.01T0
0.1T0
1T0
10T0
100T0
1000T0
k=0.5ωS(k,ω)
ω
Oscillatory motion
Jump motion
![Page 9: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/9.jpg)
The dynamical structure factor of Model 2ωS(k,ω)
ω
k=0.5
0.01T0
0.1T0
0.3T0
10T0
1000T0
Jump motion
Oscillatory motion
![Page 10: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/10.jpg)
The dynamical structure factor of Model 3k=0.5
ωS(k,ω)
ω
100T0
10T0
1T0
0.3T00.1T0
0.01T0
Jump motion
Oscillatory motion
![Page 11: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/11.jpg)
T/1
Characteristic time scales
![Page 12: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/12.jpg)
Phenomenology
Fundamental Theory
Dynamics
Single particle: Gaussian to non-Gaussian transitionSlow and fast relaxations
Specific heat: Annealed to quenched transition
Thermodynamics
Cooling-rate dependence
Construction of free energy landscape
Dynamics
Thermodynamics
Unified Theory for Glass Transition
![Page 13: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/13.jpg)
))()()(()(2
1
))(()(
log)(][)]([
212121 ll
ll
l
cdd
dd
rrrrrr
rrr
rrr
: Direct correlation function)(rc
Ramakrishnan-Yussouff free energy functional
}){,(])(exp[)( 2i
iiC RRrr
})]{,([})({ ii RR as a function of }{ iR
Free energy landscape
![Page 14: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/14.jpg)
No of atoms in the core : 32555.0 362
String motion and CRR
![Page 15: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/15.jpg)
Simultaneously and cooperatively rearranging regions
SRR: Difference between two adjacent basins
CRR: Atoms involved in the transition state
108
523.0
N
return
![Page 16: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/16.jpg)
Phenomenological understanding : Heat capacity
T. Tao &T.O(PRE 2002),T.O et al (JCP 2002),T. Tao et al (JCP2005)
aE
),( tTPa
Energy of basin a
Probability of being in basin a at t
),(),( tTPEtTE aa
a
0
0000
),(),(),(
TT
tTEtTEttTC
)0,(TC
),( TC
: Quenched
: Annealed
a
![Page 17: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/17.jpg)
)10,10,10( 642coolCt
)10( 2heatCt
Annealed-to-quenched transition and cooling rate dependence
• 20 basins:Einstein oscillators
slow
fast
T. Tao, T. O and A. Yoshimori: JCP 122, 044505 (2005)
return
![Page 18: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/18.jpg)
Trapping Diffusion ModelTrapping Diffusion Model
return
)2()( tt
)(
)()(
gcg
gcgc
TsT
TsTTTs
Waiting time distribution for jump motion
![Page 19: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/19.jpg)
Unifying concept
0t1 0T 0)( 0 Tsc
2t xT10 gTt 0D
2)(
)(
0
0
TT
TT
TsT
TsT
g
X
gcg
XcX
)/(
12
0TTT
T
gg
x
Characteristic Temperature Equation
![Page 20: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/20.jpg)
Characteristic Temperature Equation
V B Kokshenev & P D Borges, JCP 122, 114510 (2005)
g
C
T
T
0/TTg
g
C
T
T
0/TTg
return
![Page 21: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/21.jpg)
Waiting time distribution for slow relaxation
g
dgCgp0
])(exp[)()( Prob. of activation free energy
2)( tt
)(
)()(1
)(*
gcg
gcgcc
TsT
TsTTTs
S
TkTs
Waiting time distribution
gneww 0
)(/* TsSn c :Size of CRR by Adam and Gibbs
SRR
CRR
return
![Page 22: T. Odagaki and T. Ekimoto Department of Physics, Kyushu University Ngai Fest September 16, 2006](https://reader034.vdocuments.site/reader034/viewer/2022051416/56649ebd5503460f94bc667c/html5/thumbnails/22.jpg)
Non-Gaussian parameter Susceptibility
return